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A216162 Sequences A006452 and A216134 interlaced, where A216134 are the indices of the Sophie Germain triangular numbers. 10
1, 0, 1, 1, 2, 4, 4, 9, 11, 26, 23, 55, 64, 154, 134, 323, 373, 900, 781, 1885, 2174, 5248, 4552, 10989, 12671, 30590, 26531, 64051, 73852, 178294, 154634, 373319, 430441, 1039176, 901273, 2175865, 2508794, 6056764, 5253004, 12681873, 14622323, 35301410 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Recall that the Pell numbers are sequence A000129. Then (a(2n) + a(2n - 1)) - (a(2n - 2) + a(2n - 3)) = A000129(n); n>1.

It follows that sqrt(2) = lim n --> infinity ((a(2n + 2) + a(2n + 1)) - (a(2n - 2) + a(2n - 3)))/((a(2n + 2) + a(2n + 1)) - (a(2n) + a(2n - 1))).

For example, for n = 5, then ((64 + 55) - (11 + 9))/((64 + 55) - (23 + 26)) = (119 - 20)/(119 - 49) = 99/70 = 1.41428571... (accurate to 5 digits).

Observation: Bearing in mind a) that the covering radius of the Leech Lattice (Lambda_24) is sqrt 2 * the packing radius (see Conway and Sloane p. 480), and b) that pronic numbers (n^2 + n) are Kissing numbers for the A_n series of lattices (see Conway and Sloane p. 109), there appears to be a curious relationship between, at the least, the first 14 elements of this sequence and, to dimension 24, laminated lattice Kissing numbers (A002336) of form Lambda_n = n*y = n*(x^2 + x); true for n = 0, 1, 4, 6, 8, 15, 20, 24. This relationship can be segmented into two broad categories: type I and type II.

Type I

Formula: Lambda_d = d*((a(k) - a(|k-2|))^2 + (a(k) - a(|k-2|))) = d*(x^2 + x) = d*y

{k} = {0, 1, 2, 3, 4, 5, 6, 7, 10, 11}

Dimension = {d} = {0, 0, 0, 1, 1, 6, 4, 8, 15, 20}

Avg. number spheres/dimension = {y} = {0, 0, 0, 2, 2, 12, 6, 30, 156, 870}

(a(0) - a(2)) = (1 - 1) = 0 & 0*(0^2 + (0)) = 0 = Lambda_0

(a(1) - a(1)) = (0 - 0) = 0 & 0*(0^2 + (0)) = 0 = Lambda_0

(a(2) - a(0)) = (1 - 1) = 0 & 0*(0^2 + (0)) = 0 = Lambda_0

(a(3) - a(1)) = (1 - 0) = 1 & 1*(1^2 + (1)) = 2 = Lambda_1

(a(4) - a(2)) = (2 - 1) = 1 & 1*(1^2 + (1)) = 2 = Lambda_1

(a(5) - a(3)) = (4 - 1) = 3 & 6*(3^2 + (3)) = 72 = Lambda_6

(a(6) - a(4)) = (4 - 2) = 2 & 4*(2^2 + (2)) = 24 = Lambda_4

(a(7) - a(5)) = (9 - 4) = 5 & 8*(5^2 + (5)) = 240 = Lambda_8

(a(10) - a(8)) = (23 - 11) = 12 & 15*(12^2 + (12)) = 2340 = Lambda_15

(a(11) - a(9)) = (55 - 26) = 29 & 20*(29^2 + (29)) = 17400 = Lambda_20

Type II

Formula: Lambda_d' = d'*((a(k') - a(|k'-1|))^2 + (a(k') - a(k'-1))) = d'*(x'^2 + x') = d'*y'

{k'} = {0, 1, 3, 5, 7, 13}

Dimension = {d'} = {1, 0, 0, 4, 8, 24}

Avg. number spheres/dimension = {y'} = {2, 0, 0, 6, 30, 8190}

Note that (y' + 2) is a power of 2 and, therefore, y'/2 is a Ramanujan-Nagell triangular number (A076046; also see A215929).

(a(0) - a(1)) = (1 - 0) = 1 & 1*(1^2 + (1)) = 2 = Lambda_1

(a(1) - a(0)) = (0 - 1) = -1 & 0*(-1^2 + (-1)) = 0 = Lambda_0

(a(3) - a(2)) = (1 - 1) = 0 & 0*(0^2 + (0)) = 0 = Lambda_0

(a(5) - a(4)) = (4 - 2) = 2 & 4*(2^2 + (2)) = 24 = Lambda_4

(a(7) - a(6)) = (9 - 4) = 5 & 8*(5^2 + (5)) = 240 = Lambda_8

(a(13) - a(12)) = (154 - 64) = 90 & 24*(90^2 + (90)) = 196560 = Lambda_24

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 3rd, 1999.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,1,0,6,0,-6,0,-1,0,1)

FORMULA

A006452 alternating with A216134.

G.f. ( -1-x^3+5*x^4-3*x^5-2*x^6+x^7-2*x^8+x^9 ) / ( (x-1)*(1+x)*(x^4-2*x^2-1)*(x^4+2*x^2-1) ). - R. J. Mathar, Sep 08 2012

Given a(0)=1, a(2)=1, a(4)=2, a(6)=4, this entire sequence can be generated recursively: a(2n) = 6(a(n - 4)) - a(n - 8); n > 3, and a(2n+1) = a(2n) + (sqrt (8*a(2n) - 7) - 1)/2; for all n. - Raphie Frank, Sep 09 2012

PROG

(PARI) Vec((-1-x^3+5*x^4-3*x^5-2*x^6+x^7-2*x^8+x^9)/((x-1)*(1+x)*(x^4-2*x^2-1)*(x^4+2*x^2-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2015

CROSSREFS

Cf. A000129.

For some k in n:

a(2n) = A006452 (k^2 - 1 is triangular).

a(2n + 1) = A216134 (T_k and 2T_k + 1 are triangular).

a(2n + 1) - a(2n) = A006451 (T_k + 1 is square).

a(2n + 1) + a(2n) = A124124 (T_k and (T_k - 1)/2 are triangular).

a(4n + 1) + a(4n + 2) = A001108 (T_k is square).

a(4n + 3) + a(4n + 4) = A001652 (T_k and 2T_k are triangular).

Sum(a(n)) - 1 = A048776 for even n (the second partial summation of the Pell numbers).

Sequence in context: A272196 A335057 A039887 * A114215 A292302 A151712

Adjacent sequences:  A216159 A216160 A216161 * A216163 A216164 A216165

KEYWORD

nonn,easy

AUTHOR

Raphie Frank, Sep 07 2012

STATUS

approved

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Last modified May 26 02:56 EDT 2020. Contains 334613 sequences. (Running on oeis4.)