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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

(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).

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

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

EXTENSIONS

Edited by N. J. A. Sloane, May 24 2021

STATUS

approved

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Last modified June 15 04:47 EDT 2021. Contains 345043 sequences. (Running on oeis4.)