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A051799 Partial sums of A007587. 4
1, 14, 60, 170, 385, 756, 1344, 2220, 3465, 5170, 7436, 10374, 14105, 18760, 24480, 31416, 39729, 49590, 61180, 74690, 90321, 108284, 128800, 152100, 178425, 208026, 241164, 278110, 319145, 364560, 414656, 469744, 530145, 596190 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

4-dimensional pyramidal number, composed of consecutive 3-dimensional slices; each of which is a 3-dimensional 12-gonal (or dodecagonal) pyramidal number; which in turn is composed of consecutive 2-dimensional slices 12-gonal numbers. - Jonathan Vos Post, Mar 17 2006

Convolution of A000027 with A051624 (excluding 0). - Bruno Berselli, Dec 07 2012

REFERENCES

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.

Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

LINKS

Table of n, a(n) for n=0..33.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Index to sequences related to pyramidal numbers.

FORMULA

a(n) = C(n+3, 3)*(5*n+2)/2 = (n+1)*(n+2)*(n+3)*(5*n+2)/12.

G.f.: (1+9*x)/(1-x)^5.

From Amiram Eldar, Feb 11 2022: (Start)

Sum_{n>=0} 1/a(n) = (125*log(5) + 10*sqrt(5*(5-2*sqrt(5)))*Pi - 50*sqrt(5)*log(phi) - 84)/104, where phi is the golden ratio (A001622).

Sum_{n>=0} (-1)^n/a(n) = (50*sqrt(5)*log(phi) + 5*sqrt(50-10*sqrt(5))*Pi - 256*log(2) + 90)/52. (End)

MATHEMATICA

Accumulate[Table[n(n+1)(10n-7)/6, {n, 0, 50}]] (* Harvey P. Dale, Nov 13 2013 *)

PROG

(Magma) /* A000027 convolved with A051624 (excluding 0): */ A051624:=func<n | n*(5*n-4)>; [&+[(n-i+1)*A051624(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012

CROSSREFS

Cf. A007587, A001622, A051624.

Cf. A093645 ((10, 1) Pascal, column m=4).

Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

Sequence in context: A158058 A100171 A063492 * A164540 A140184 A264854

Adjacent sequences:  A051796 A051797 A051798 * A051800 A051801 A051802

KEYWORD

nonn,easy

AUTHOR

Barry E. Williams, Dec 11 1999

STATUS

approved

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Last modified June 29 07:33 EDT 2022. Contains 354910 sequences. (Running on oeis4.)