OFFSET
1,2
COMMENTS
According to Dickson, Pollock conjectures that a(n) <= 5 for all n. Watson shows that a(n) <= 8 for all n, and Salzer and Levine show that a(n) <= 5 for n <= 452479659. - N. J. A. Sloane, Jul 15 2011
Possible correction of the first comment by Sloane 2011: it appears to me from the linked reference by Salzer and Levine 1968 that 452479659 is instead the upper limit for sums of five Qx = Tx + x, where Tx are the tetrahedral numbers we want. They also mention an upper limit for sums of five Tx, which is: a(n) <= 5 for n <= 276976383. - Ewoud Dronkert, May 30 2024
If we use the greedy algorithm for this, we get A281367. - N. J. A. Sloane, Jan 30 2017
Could be extended with a(0) = 0, in analogy to A061336. Kim (2003, first row of table "d = 3" on p. 73) gives max {a(n)} = 5 as a "numerical result", but the value has no "* denoting exact values" (see Remark at end of paper), which means this could be incorrect. - M. F. Hasler, Mar 06 2017, edited Sep 22 2022
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 436.
L. E. Dickson, History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 1952, see p. 13.
LINKS
Lars Blomberg, Table of n, a(n) for n = 1..10000
Hyun Kwang Kim, On regular polytope numbers, Proc. Amer. Math. Soc. 131 (2003), pp. 65-75.
F. Pollock, On the Extension of the Principle of Fermat's Theorem of the Polygonal Numbers to the Higher Orders of Series Whose Ultimate Differences Are Constant. With a New Theorem Proposed, Applicable to All the Orders, Abs. Papers Commun. Roy. Soc. London 5, 922-924, 1843-1850.
H. E. Salzer and N. Levine, Table of Integers Not Exceeding 1000000 that are Not Expressible as the Sum of Four Tetrahedral Numbers, Math. Comp. 12, 141-144, 1958.
H. E. Salzer and N. Levine, Proof that every integer <= 452,479,659 is a sum of five numbers of the form Q_x = (x^3+5x)/6, x >= 0, Math. Comp., (1968), 191-192.
N. J. A. Sloane, Transforms.
G. L. Watson, Sums of eight values of a cubic polynomial, J. London Math. Soc., 27 (1952), 217-224.
Eric Weisstein's World of Mathematics, Tetrahedral Number.
MAPLE
tet:=[seq((n^3-n)/6, n=1..20)];
LAGRANGE(tet, 8, 120); # the LAGRANGE transform of a sequence is defined in A193101. - N. J. A. Sloane, Jul 15 2011
# Alternative:
N := 10000:
L := [seq(0, i=1..N)] :
# put 1's where tetrahedral numbers reside
for i from 1 to N do
Aj := A000292(i) ;
if Aj <= N then
L := subsop(Aj=1, L) ;
end if;
end do:
for a from 1 do
# select positions of a's, skip forward by all available Aj and
# if that addresses a not-yet-set position in the array put a+1 there.
for i from 1 to N do
if op(i, L) =a then
for j from 1 do
Aj := A000292(j) ;
if i+Aj <=N and op(i+Aj, L) = 0 then
L := subsop(i+Aj=a+1, L) ;
end if;
if i +Aj > N then
break ;
end if;
end do:
end if;
end do:
# if all L[] are non-zero, terminate the loop
allset := true;
for i from 1 to N do
if op(i, L) = 0 then
allset := false ;
break ;
end if;
end do:
if allset then
break ;
end if;
end do:
seq( L[i], i=1..N) ; # R. J. Mathar, Jun 06 2025
PROG
(PARI) \\ available on request. - M. F. Hasler, Mar 06 2017
(PARI)
seq(N) = {
my(a = vector(N, k, 8), T = k->(k*(k+1)*(k+2))\6);
for (n = 1, N,
my (k1 = sqrtnint((6*n)\8, 3), k2 = sqrtnint(6*n, 3));
while(n < T(k2), k2--); if (n == T(k2), a[n] = 1; next());
for (k = k1, k2, a[n] = min(a[n], a[n - T(k)] + 1))); a;
};
seq(102) \\ Gheorghe Coserea, Mar 14 2017
CROSSREFS
Cf. A000292 (tetrahedral numbers, indices of 1s), A102795 (indices of 2s), A102796 (indices of 3s), A102797 (indices of 4s), A000797 (numbers that need 5 tetrahedral numbers).
See also A102798-A102806, A102855-A102858, A193101, A193105, A281367 (the "triangular nachos" numbers).
Cf. A061336 (analog for triangular numbers).
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Feb 26 2005
EXTENSIONS
Edited by N. J. A. Sloane, Jul 15 2011
Edited by M. F. Hasler, Mar 06 2017
STATUS
approved